Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $x = \dfrac{-8}{6(5q - 3)} \div \dfrac{8q}{4q(5q - 3)} $
Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{-8}{6(5q - 3)} \times \dfrac{4q(5q - 3)}{8q} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ -8 \times 4q(5q - 3) } { 6(5q - 3) \times 8q } $ $ x = \dfrac{-32q(5q - 3)}{48q(5q - 3)} $ We can cancel the $5q - 3$ so long as $5q - 3 \neq 0$ Therefore $q \neq \dfrac{3}{5}$ $x = \dfrac{-32q \cancel{(5q - 3})}{48q \cancel{(5q - 3)}} = -\dfrac{32q}{48q} = -\dfrac{2}{3} $